Quadrature Schemes for the Integration of the Free-Surface Green’s Function Within the Open-Source Boundary Element Method Library BEMUse

Abstract The direct boundary element method has a computational expense which scales quadratically with the number of panel elements. This is a significant bottleneck in the application of the method to large systems. As a preliminary step towards the implementation of a solver with linear computational expense, a detailed study is carried out to investigate quadrature schemes for the integration of the free-surface Green’s function over hydrodynamic panels in the frequency domain. Important parameters are investigated including quadrature order, quadrature rule and panel refinement. Integration errors are quantified by comparison to analytical expressions and higher-order quadrature schemes. Important observations relevant to the removal of irregular frequencies are observed for practical implementation into a boundary element method solver. The order of the method is demonstrated and the influence of the quadrature order on this scaling is quantified. The schemes have been implemented within the open-source boundary element code BEMUse. The ability of the quadrature schemes to correctly calculate the surface hydrodynamic potential is validated by comparing to analytical results for a submerged, oscillating hemisphere. The schemes are verified for realistic offshore wind energy floating platforms geometries by comparison to validated software. It is seen that the quadrature methods allow for accurate calculation of added mass, damping coefficients and excitation forces.